Quarterly of Applied Mathematics

Quarterly of Applied Mathematics

Online ISSN 1552-4485; Print ISSN 0033-569X



Stochastic Radon operators in porous media hydrodynamics

Authors: George Christakos and Dionissios T. Hristopulos
Journal: Quart. Appl. Math. 55 (1997), 89-112
MSC: Primary 76S05; Secondary 60G60, 76M35, 86A05
DOI: https://doi.org/10.1090/qam/1433754
MathSciNet review: MR1433754
Full-text PDF Free Access

Abstract | References | Similar Articles | Additional Information

Abstract: A space transformation approach is established to study partial differential equations with space-dependent coefficients modelling porous media hydrodynamics. The approach reduces the original multi-dimensional problem to the one-dimensional space and is developed on the basis of Radon and Hilbert operators and generalized functions. In particular, the approach involves a generalized spectral decomposition that allows the derivation of space transformations of random field products. A Plancherel representation highlights the fact that the space transformation of the product of random fields inherently contains integration over a ``dummy'' hyperplane. Space transformation is first examined by means of a test problem, where the results are compared with the exact solutions obtained by a standard partial differential equation method. Then, exact solutions for the flow head potential in a heterogeneous porous medium are derived. The stochastic partial differential equation describing three-dimensional porous media hydrodynamics is reduced into a one-dimensional integro-differential equation involving the generalized space transformation of the head potential. Under certain conditions the latter can be further simplified to yield a first-order ordinary differential equation. Space transformation solutions for the head potential are compared with local solutions in the neighborhood of an expansion point which are derived by using finite-order Taylor series expansions of the hydraulic log-conductivity.

References [Enhancements On Off] (What's this?)

  • [1] I. Bialynicki-Birula, M. Cieplak, and J. Kaminski, Theory of Quanta, Oxford University Press, New York, N. Y., 1992
  • [2] G. Christakos, The space transformations and their applications in systems modelling and simulation, Proc. 12th Intern. Confer, on Modelling and Simulation (AMSE) 1 (3), Athens, Greece, 1984, pp. 49-68
  • [3] G. Christakos, Random Field Models in Earth Sciences, Academic Press, San Diego, CA, 1992
  • [4] G. Christakos and D. T. Hristopulos, Stochastic space transformation techniques in subsurface hydrology-Part 2: Generalized spectral decompositions and Plancherel representations, Stochastic Hydrology and Hydraulics 8, no. 2, 117-138 (1994)
  • [5] R. Courant and D. Hilbert, Methods of Mathematical Physics, Vol. I, J. Wiley, New York, N. Y., 1953 MR 0065391
  • [6] I. M. Gel'fand and G. E. Shilov, Generalized Functions, Vol. 1, Academic Press, New York, N. Y., 1964 MR 0166596
  • [7] S. Helgason, The Radon Transform, Birkhauser, Boston, Basel, Stuttgart, 1980 MR 573446
  • [8] F. John, Plane Waves and Spherical Means, Springer-Verlag, New York, N. Y., 1955
  • [9] M. Loeve, Probability Theory, Van Nostrand, Princeton, 1953 MR 0203748
  • [10] A. Scheidegger, Physics of Flow Through Porous Media, University of Toronto Press, Toronto, Canada, 1960 MR 0127717
  • [11] A. K. Jain, Fundamentals of Digital Image Processing, Prentice Hall, Englewood Cliffs, NJ, 1989

Similar Articles

Retrieve articles in Quarterly of Applied Mathematics with MSC: 76S05, 60G60, 76M35, 86A05

Retrieve articles in all journals with MSC: 76S05, 60G60, 76M35, 86A05

Additional Information

DOI: https://doi.org/10.1090/qam/1433754
Article copyright: © Copyright 1997 American Mathematical Society

American Mathematical Society